Asymptotics for the ruin time of a piecewise exponential Markov process with jumps
AAsymptotics for the ruin time of a piecewiseexponential Markov process with jumps
Anders Rønn-Nielsen
Department of Mathematical Sciences, University of Copenhagen
Abstract
In this paper a class of Ornstein–Uhlenbeck processes driven by com-pound Poisson processes is considered. The jumps arrive with exponentialwaiting times and are allowed to be two-sided. The jumps are assumed toform an iid sequence with distribution a mixture (not necessarily convex)of exponential distributions, independent of everything else. The fact thatdownward jumps are allowed makes passage of a given lower level possi-ble both by continuity and by a jump. The time of this passage and thepossible undershoot (in the jump case) is considered. By finding partialeigenfunctions for the infinitesimal generator of the process, an expressionfor the joint Laplace transform of the passage time and the undershootcan be found.From the Laplace transform the ruin probability of ever crossing thelevel can be derived. When the drift is negative this probability is less thanone and its asymptotic behaviour when the initial state of the process tendsto infinity is determined explicitly.The situation where the level to cross decreases to minus infinity ismore involved: The level to cross plays a much more fundamental role inthe expression for the joint Laplace transform than the initial state of theprocess. The limit of the ruin probability in the positive drift case andthe limit of the distribution of the undershoot in the negative drift case isderived.
Keywords:
Asymptotic ruin probabilities; Integration contour; Ornstein–Uhlenbeck process; Partial Eigenfunction; Shot–noise process
The main aim of this paper is to determine the asymptotic behaviour of theruin probability for a certain class of time–homogeneous Markov processes withjumps. These processes, referred to as X below, can be viewed as Ornstein–Uhlenbeck processes satisfyingd X t = κX t d t + d U t , (1)driven by a compound Poisson process ( U t ). The ruin time, τ ( (cid:96) ), is defined asthe time to passage below (cid:96) for an initial state x > (cid:96) . The passage below (cid:96) can be a result of a downward jump, and in some cases a continuous passagethrough (cid:96) is is also possible. The main results give asymptotic descriptions of1 a r X i v : . [ m a t h . P R ] A ug x ( τ ( (cid:96) ) < ∞ ), when κ > x → ∞ and (cid:96) → −∞ . Furthermore, thelimit distribution of the undershoot in case of passage by jump is determinedfor κ < (cid:96) → −∞ .It will be assumed that the driving compound Poisson process has a specialjump structure. Both the downward and upward jumps are assumed to have adensity (not the same) that is a linear – not necessarily convex – combinationof exponential densitiesIt is important to distinguish between two different scenarios: Whether thedrift κ is positive, hence X is transient, or the drift is negative, in which case theprocess X is recurrent. In the negative drift case the probability P x ( τ ( (cid:96) ) < ∞ )(with τ ( (cid:96) ) denoting the time of passage) of ever crossing below (cid:96) when startingat x is always 1. When the drift κ is positive we have that P x ( τ ( (cid:96) ) < ∞ ) < x → ∞ or (cid:96) → −∞ .The distribution of the passage time (and by that also the ruin probability)is determined through the Laplace transform. This is found by exploiting cer-tain stopped martingales derived from using bounded partial eigenfunctions forthe infinitesimal generator for X . An explicit expression for the Laplace trans-form is determined in [10]. Here the partial eigenfunctions are found as linearcombinations of functions given by contour integrals in the complex plane. Alsothe Laplace transform ends up being a linear combination of these integrals.It is the resulting Laplace transform from [10] that we shall investigate in thispaper.In the present paper the asymptotics of P x ( τ ( (cid:96) ) < ∞ ) is explored in bothof the situations x → ∞ and (cid:96) → −∞ . This becomes a question about findingthe asymptotics for the complex contour integrals mentioned above. It turnsout that the (cid:96) → −∞ problem is the far most complicated because the depen-dence of (cid:96) in the construction of the partial eigenfunctions is more involved.Nevertheless, the need of exploring the asymptotic behaviour of the integralsis similar. When x → ∞ we see that P x ( τ ( (cid:96) ) < ∞ ) decreases exponentially(adjusted by some specified power function) with the exponential parameterfrom the leading exponential part of the downward jumps.The technique of using partial eigenfunctions for the infinitesimal generatorhas appeared before. Paulsen and Gjessing, [13], considers a model like thepresent, but in the more general (and also different) setup dX t = ( p + κX t ) dt − dU t + (cid:113) σ + σ X t dB t + X t d ˜ U t . (2)Here both U and ˜ U are compound Poisson processes of the form (cid:80) N t n =1 V n . In[13] it is shown that a partial eigenfunction for the corresponding infinitesimalgenerator for (2) will lead to the ruin probability and also the Laplace transformfor the ruin time. [5] shows in a model without σ and ˜ U the existence ofthis partial eigenfunction under some smoothness assumptions about the jumpdistributions in U . This result is extended to weaker assumptions in [6].In the case of σ = σ = 0, without ˜ U , and assuming exponential negativejump and no positive jumps, an explicit formula for the Laplace transform isdetermined in [13]. Furthermore, the exponential decrease in P x ( τ ( (cid:96) ) < ∞ ) is2erived in the x → ∞ asymptotic situation for some fixed 0 < (cid:96) < x . For thecase of exponential negative jumps also see Asmussen [1], Chapter VII.In the present paper the jump distributions are assumed to be light tailed.The existing literature does not contain very explicit results for the asymptoticruin probability with that kind of jump distributions. In [4] and [14] it is provedin the σ = 0 case with κ = sup { a | E [ e aU ] < ∞} that for any (cid:15) > x →∞ e ( κ − (cid:15) ) x P x ( τ (cid:96) < ∞ ) = 0 and lim x →∞ e ( κ + (cid:15) ) x P x ( τ (cid:96) < ∞ ) = ∞ . In the case of heavy tailed jump distributions there are more explicit results forthe asymptotic behaviour of the ruin probability. In [11] results are obtainedfor the asymptotics of the finite horizon ruin probability P x ( τ ( (cid:96) ) ≤ T ) in afairly general model with σ = 0 and subexpontial jump distributions. Similarresults are reached in [3] in the infinite horizon case. Here the jumps belong toa less general class of heavy tailed distributions.In [7], [8], [9] the following model class of certain Markov modulated L´evyprocesses X t = x + (cid:90) t β J s d s + (cid:90) t σ J s − d B s − N t (cid:88) n =1 U n is studied. The same partial eigenfunction technique is applied, and it is showedthat the partial eigenfunctions (and thereby also the ruin probabilities) can beexpressed as a linear combination of exponential functions (evaluated in thestarting point x ). Hence, the asymptotic behaviour of the probability when x → ∞ reduces to finding the exponential function with the slowest decrease.Since the model is additive, the level (cid:96) that is to be crossed at the time of ruin,enters into the setup symmetric to x . Hence, the asymptotics when (cid:96) → −∞ are just as easy to derive. In Novikov et. al, [12], the Laplace transform isdetermined for a shot–noise model with exponentially distributed downwardjumps (and no positive jumps allowed) for a process with negative drift. TheLaplace transform was also derived in the case of uniformly distributed down-ward jumps. In [2] these results are extended to a more general driving L´evyprocess instead of a compound Poisson process. In [2, 12] some asymptoticresults for the distribution of τ ( (cid:96) ) are carried out. Here the limit distributionof τ ( (cid:96) ) is expressed when (cid:96) → −∞ for some fixed starting point x and negativedrift. This is a limit that is not considered in the present paper.The paper is organised as follows. In Section 2 the setup is defined andthe relevant results from [10] reproduced. Theorem 2.1 is also reformulated ina different (and appearently more complicated) version as Theorem 2.2 thatturns out to fit the asymptotic considerations better. In Section 2.1 the choiceof some complex integration contours that are applied in Theorem 2.1 andTheorem 2.2 is discussed. This choice differs from the proposed contours in [10]in order to suit the further calculations. In Section 3 the asymptotic behaviourof P x ( τ ( (cid:96) ) < ∞ ) is expressed when x → ∞ and in Section 4 the limit when (cid:96) → −∞ is found. Finally the limit of the distribution of the undershoot isexpressed for the negative drift case when (cid:96) → −∞ .3 The model and previous results
Consider a process X with state space R defined by the following stochasticdifferential equation: d X t = κX t d t + d U t , (3)where ( U t ) is a compound Poisson process defined by U t = N t (cid:88) n =1 V n . (4)Here ( V n ) are iid with distribution G and ( N t ) is a Poisson process with param-eter λ . Both the downward and the upward part of the jump distribution G isassumed to be a linear combination of exponential distributions. We use thedecomposition G = pG − + qG + where 0 < p ≤ q = 1 − p , G − is restricted to R − = ( −∞ ; 0) and G + is restricted to R + = (0; ∞ ). That is, G − ( du ) = g − ( u ) du = r (cid:88) k =1 α k µ k e µ k u for u < G + ( du ) = g + ( u ) du = s (cid:88) d =1 β d ν d e − ν d u for u > . (5)The distribution parameters are arranged such that 0 < µ < · · · < µ r , 0 <ν < · · · < ν s and α i , β j (cid:54) = 0. Since g − and g + need to be densities (cid:80) α i = 1and (cid:80) β j = 1. Furthermore both α > β >
0. The remaining densityparameters are not necessarily non–negative.Between jumps the solution process X behaves deterministically following anexponential function. Assume x > P x for the probability space,where X = x P x –almost surely. Let E x be the corresponding expectation.Define for (cid:96) < x the stopping time τ by τ = τ ( (cid:96) ) = inf { t > X t ≤ (cid:96) } . (6)For ease of notation (cid:96) is most often suppressed. Furthermore define the under-shoot Z = (cid:96) − X τ , (7)which is well–defined on the set { τ < ∞} . Note that the level (cid:96) can by crossedthrough continuity as well as a result of a downward jump. Of interest is a jointexpression about ( τ < ∞ ) and the distribution of Z . This is expressed throughthe expressions E x [ e − ζZ ; A j ] and E x [ A c ] , (8)where A j and A c is a partition of the set { τ < ∞} into the jump case A j = { τ < ∞ , X τ < (cid:96) } and the continuity case A c = { τ < ∞ , X τ = (cid:96) } . The expressions in(8) can be found from solving two equations E x [e − ζZ ; A j ] + f i ( (cid:96) ) E x [ A c ] = f i ( x ) , i = 1 , , (9)4here f and f are partial eigenfunctions for the infinitesimal generator A forthe process: f i : R → C are bounded and differentiable on [ (cid:96) ; ∞ ) and satisfythe condition that A f i ( x ) = 0 for all x ∈ [ (cid:96) ; ∞ ) , where A is defined by A f ( x ) = κxf (cid:48) ( x ) + λ (cid:90) R (cid:16) f ( x + y ) − f ( x ) (cid:17) G (d y ) , (10)for details, see [10]. In addition, each f i has the following exponential form onthe interval ( −∞ ; (cid:96) ) f i ( x ) = e − ζ ( (cid:96) − x ) for x < (cid:96) . It is important to notice that there exists some situations where only one partialeigenfunction is needed: If (cid:96)κ > P x ( A c ) of crossing (cid:96) throughcontinuity is 0 (recall that the process is deterministic and monotone betweenjumps). In this case finding E x [e − ζZ ; A j ] is even simpler (from (9) with the A c part equal 0): E x [e − ζZ ; A j ] = f ( x ) , (11)where f is the single partial eigenfunction.In the negative drift case ( κ <
0) the recurrence of X gives that P x ( A j ) + P x ( A c ) = P x ( τ < ∞ ) = 1. If furthermore ζ = 0 the desired expressions in(8) reduce to the probabilities P x ( A j ) and P x ( A c ). Hence, only one partialeigenfunction is needed in order to solve the equation.In [10, Theorem 4] a result is given that sketches how to construct suchpartial eigenfunctions. In the following this theorem is reformulated in order tofit the further calculations. Define f ( y ) = (cid:26) y ≥ (cid:96) e − ζ ( (cid:96) − y ) y < (cid:96) , (12)and f Γ ( y ) = (cid:26) (cid:82) Γ ψ ( z )e − yz d z y ≥ (cid:96) y < (cid:96) , (13)where ψ is the complex valued kernel defined by ψ ( z ) = z − (cid:32) r (cid:89) k =1 ( z − µ k ) − pλα k κ (cid:33)(cid:32) s (cid:89) d =1 ( z + ν d ) − qλβ d κ (cid:33) , (14)and Γ is some suitable curve in the complex plane of the form Γ = { γ ( t ) : δ
00 otherwise . (18)For convenience we shall use the following definitions6 efinition 2.1. M k Γ i = (cid:90) Γ i ψ ( z ) z − µ k d z i = 1 , . . . , m, k = 1 , . . . , rM d Γ i = (cid:90) Γ i ψ ( z ) ν d + z d z i = 1 , . . . , m, d = 1 , . . . , sN k Γ j = (cid:90) Γ j ψ ( z ) µ k − z d z j = 1 , . . . , n, k = 1 , . . . , rN d Γ j = (cid:90) Γ j ψ ( z ) ν d + z d z j = 1 , . . . , n, d = 1 , . . . , sN k Γ j = (cid:90) Γ j ψ ( z ) z − µ k e − (cid:96)z d z j = 1 , . . . , n, k = 1 , . . . , r . We will need
Condition 2.1.
Let ζ ≥ be given and let f , f i and f j be defined asin (18) for i = 1 , . . . , m and j = 1 , . . . , n such that all Γ i ⊂ C + and Γ j ⊂ C are suitable complex curves ( ψ should have holomorfic versions containing thesecurves). Assume for ψ and Γ i , i = 1 , . . . , m , that(i) (cid:82) Γ i | ψ ( z ) | d z < ∞ (ii) (cid:82) Γ i | ψ ( z ) | | z | e − y Re z d z < ∞ for all y > (iii) (cid:82) Γ i | ψ ( z ) z − µ k | d z < ∞ for k = 1 , . . . , r (iv) (cid:82) Γ i | ψ ( z ) z + ν d | d z < ∞ for d = 1 , . . . , s (v) ψ ( γ i ( δ i )) γ i ( δ i )e − yγ i ( δ i ) = ψ ( γ i ( δ i )) γ i ( δ i )e − yγ i ( δ i ) for all y > ,and similarly for ψ and Γ j that(i’) (cid:82) Γ j | ψ ( z ) | d z < ∞ (ii’) (cid:82) Γ j | ψ ( z ) | e − (cid:96) Re z d z < ∞ (iii’) (cid:82) Γ j | ψ ( z ) | | z | e − y Re z d z < ∞ for all y ∈ [ (cid:96) ; 0[ (iv’) (cid:82) Γ j | ψ ( z ) z − µ k | d z < ∞ for k = 1 , . . . , r (v’) (cid:82) Γ j | ψ ( z ) z − µ k | e − (cid:96)z d z < ∞ for k = 1 , . . . , r (vi’) (cid:82) Γ j | ψ ( z ) z + ν d | d z < ∞ for d = 1 , . . . , s (vii’) ψ ( γ j ( δ j )) γ j ( δ j )e − yγ ( δ j ) = ψ ( γ j ( δ j )) γ j ( δ j )e − yγ j ( δ j ) for all (cid:96) ≤ y < .for j = 1 , . . . , n . Theorem 2.2.
Assume that the integration contours Γ i , i = 1 , . . . , m and Γ j , j = 1 , . . . , n satisfy the conditions in Condition 2.1. Define f : R → C by f ( y ) = m (cid:88) i =1 c i f i ( y ) + n (cid:88) j =1 b j f j ( y ) + f ( y ) . (19) Then f is bounded and differentiable on (cid:96) → ∞ . If the constants c , . . . , c m and b , . . . , b n fulfil the equations n (cid:88) j =1 b j N k Γ j + 1 µ k + ζ = 0 (20) and (cid:32) m (cid:88) i =1 c i M k Γ i (cid:33) + (cid:32) n (cid:88) j =1 b j N k Γ j (cid:33) = 0 (21) for k = 1 , . . . , r together with (cid:32) n (cid:88) j =1 b j N d Γ j (cid:33) − (cid:32) m (cid:88) i =1 c i M d Γ j (cid:33) = 0 (22) for d = 1 , . . . , s , then f is a partial eigenfunction for A .Proof. As in the proof of [10, Theorem 4] it is seen that for y ≥ A f i = pλ r (cid:88) k =1 α k µ k M k Γ i e − µ k y and for (cid:96) ≤ y < A f i = − qλ s (cid:88) d =1 β d ν d M d Γ i e ν d y Furthermore, we find for y ≥ A f j = pλ r (cid:88) k =1 α k µ k N k Γ j e − µ k y + pλ r (cid:88) k =1 α k µ k N k Γ j e µ k (cid:96) e − µ k y and finally, for (cid:96) ≤ y < A f j = qλ s (cid:88) d =1 β d ν d N d Γ j e ν d y + pλ r (cid:88) k =1 α k µ k N k Γ j e µ k (cid:96) e − µ k y Since for all y ≥ (cid:96) A f ( y ) = λ r (cid:88) k =1 α k µ k µ k + ζ e µ k (cid:96) e − µ k y , it follows that A f ( y ) = 0 for all y ≥ (cid:96) , if the equations (20)–(22) are satisfied.8 (cid:54) (cid:114) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) µ i is a singularity for ψ . µ i µ Γ i (cid:45)(cid:54) (cid:114) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) µ i is a zero for ψ . µ i Γ i Figure 1: The contour Γ i in the two cases: µ i is a singularity (left) for ψ and µ i is a zero (right) There are several possible choices for the integration contours, see [10]. Thechoice described in the following applies to cases with positive drift κ and willdiffer from the ones defined in [10]. The situation κ < (cid:96) >
0. Then only one partial eigenfunction is neededand we shall use Theorem 2.1. The definition of the m = r contours has itsstarting point in the zeros and singularities of the kernel ψ . The real–valuedpoints − ν s , . . . , − ν , , µ , . . . , µ r from (5) are all such zeros or singularities.The contours Γ , . . . , Γ r are chosen as follows • If µ i is a zero for ψ defineΓ i = { µ i + (1 + i ) t : 0 ≤ t < ∞} . • If µ i is a singularity for ψ defineΓ i = { µ + ( − i ) t : −∞ < t ≤ } ∪ { µ + (1 + i ) t : 0 ≤ t < ∞} for a µ ∈ ( µ i − , µ i ) (with the convention µ = 0).A sketch of the chosen contours can be seen in Figure 1. Next assume that (cid:96) < , . . . , Γ r one can use Γ , . . . , Γ r from above. It remains to find n = r + s + 1 contours Γ , . . . , Γ r + s +1 , in orderto construct two eigenfunctions. For convenience let p , . . . , p r + s +1 denote thepoints − ν s , . . . , − ν , , µ , . . . , µ r and use the following recipe: • If p i is a zero for ψ defineΓ i = { p i + ( − i ) t : 0 ≤ t < ∞} If p i is a singularity for ψ defineΓ i = { p + (1 + i ) t : −∞ < t ≤ } ∪ { p + ( − i ) t : 0 ≤ t < ∞} for a p ∈ ( p i ; p i +1 ) (with the convention p r + s +2 = ∞ ). Remark 2.1.
For the contours Γ i corresponding to a singularity the specificchoice of µ in ( µ i − , µ i ) is without influence as a result of Cauchy’s Theorem.In fact, µ can be chosen freely in ( µ l , µ i ) where µ l is the largest singularity for ψ less than µ i (remember that 0 is a singularity so that µ l ≥ ). Moreover,it can never happen that f Γ i = f Γ i +1 in the case where both µ i and µ i +1 aresingularities. If µ i , the singularity that separates the two contours, is of order ρ < with ρ / ∈ Z this is secured from the use of different versions of the complexlogarithm in the respective domains of the contours. If the singularity µ i is aninteger the argument that f Γ i (cid:54) = f Γ i +1 is based on Cauchy’s Theorem. x → ∞ When the drift κ > P x ( τ < ∞ ) <
1. Furthermore, the probabilitydecreases when the initial value x increases. Solving the equation system (9)w.r.t. P x ( τ < ∞ ) = P x ( A c ) + P x ( A j ) we have for (cid:96) < P x ( τ < ∞ ) = f ( x ) 1 − f ( (cid:96) ) f ( (cid:96) ) − f ( (cid:96) ) + f ( x ) f ( (cid:96) ) − f ( (cid:96) ) − f ( (cid:96) ) , (23)where f and f are the two partial eigenfunctions constructed in Theorem 2.2.When (cid:96) > E x [ A j ] = f ( x ) , where f is the single eigenfunction constructed in Theorem 2.1. It is essentialthat the construction of the partial eigenfunctions f and f (or f in the (cid:96) > x . The behaviour of the probability P x ( τ < ∞ ) tobe studied is therefore only determined by the behaviour of the two partialeigenfunctions f and f when x → ∞ . We have the following result: Theorem 3.1.
There exists a constant K such that lim x →∞ P x ( τ < ∞ )e − µ x x − pα λκ − = K .
The constant K is expressed explicitly in (31) below when (cid:96) < and in (32)when (cid:96) > . For the later use of the results it is convenient to formulate part of theproof of Theorem 3.1 as self–contained lemmas. Furthermore, the definitions ρ j = − pα j λ/κ and ψ \{ µ j } ( z ) = z − r (cid:89) k =1 ,k (cid:54) = j ( z − µ k ) − pαkλκ (cid:32) s (cid:89) d =1 ( z + ν d ) − qβdλκ (cid:33) . j = 1 , . . . , r will be convenient. Now f j can be written as f j ( x ) = (cid:90) Γ ( z − µ j ) ρ j ψ \{ µ j } ( z )e − xz d z . The first lemma concerns the case, where α j <
0. Here µ j is a zero for ψ , andΓ j = { µ j + (1 + i ) t : 0 ≤ t < ∞} . We find Lemma 3.1.
Assume α j < . Then it holds that lim x →∞ f j ( x )e − µ j x x ρ − = ψ \{ µ j } ( µ j ) (cid:90) Γ z ρ j e − z d z , (24) where Γ is the integration contour Γ = { (1 + i ) t : 0 ≤ t < ∞} . (25) Proof.
The expression of f j ( x ) can be rewritten in the following way f j ( x ) = (cid:90) Γ j ( z − µ j ) ρ j ψ \{ µ j } ( z )e − xz d z = (cid:90) ∞ (1 + i ) (cid:0) (1 + i ) t (cid:1) ρ j ψ \{ µ j } (cid:0) µ j + (1 + i ) t (cid:1) e − x ( µ j +(1+ i ) t ) d t = x − ρ j − e − µ j x (cid:90) ∞ (1 + i ) (cid:0) (1 + i ) s (cid:1) ρ j ψ \{ µ j } (cid:0) µ j + (1 + i ) sx (cid:1) e − s (1+ i ) d s , (26)where the substitution s = tx has been used. Consider the function t (cid:55)→| ψ \{ µ j } (cid:0) µ j + (1 + i ) t (cid:1) | , which is continuous and strictly positive. Furthermoreit is O ( | µ j + (1 + 2 i ) t | − − λ/κ − ρ j ), when t → ∞ . This gives the existence of aconstant C < ∞ such that | ψ \{ µ j } (cid:0) µ j + (1 + i ) t (cid:1) | ≤ C for all t ≥ . In particular, this holds when t = s/x for all s ≥ x >
0. Thus, thefunction s (cid:55)→ C | (1 + i )((1 + 2 i ) s ) ρ j | e − s is an integrable upper bound for the integrand in the last line of (26). Bydominated convergence we get thatlim x →∞ (cid:90) ∞ (1 + i ) (cid:0) (1 + i ) s (cid:1) ρ j ψ \{ µ j } (cid:0) µ j + (1 + i ) sx (cid:1) e − s (1+ i ) d s = (cid:90) ∞ (1 + i ) (cid:0) (1 + i ) s (cid:1) ρ j ψ \{ µ j } ( µ j )e − s (1+ i )) d s = ψ \{ µ j } ( µ j ) (cid:90) Γ z ρ j e − z d z . Hence the result is shown. 11or the proof of the next lemma we defineΓ µ = { µ + ( − i ) t : −∞ < t ≤ } ∪ { µ + (1 + i ) t : 0 < t < ∞} , for µ >
0. Note that if α j >
0, then µ j is a singularity for ψ and Γ j = Γ µ ,where µ ∈ ( µ j − , µ j ). We have Lemma 3.2.
Assume that α j > . Then lim x →∞ f j ( x ) x ρ j − e − µ j x = ψ \{ µ j } ( µ j ) (cid:90) Γ − a z ρ j e − z d z , (27) where Γ − a = {− a + ( − i ) t : −∞ < t ≤ } ∪ {− a + (1 + i ) t : 0 < t < ∞} and a > is any positive real number.Proof. In Remark 2.1 it was argued that f j ( x ) = f µ (cid:48) ( x )for all µ (cid:48) ∈ ( µ l , µ j ), where µ l is the largest singularity for ψ less than µ j . Wechoose µ (cid:48) = µ j − ax for some suitable a >
0. Hence, f j ( x )= f µj − a/x ( x )= (cid:90) ∞ (1 + i ) (cid:0) − ax + (1 + i ) t (cid:1) ρ ψ \{ µ j } (cid:0) µ j − ax + (1 + i ) t (cid:1) e − xµ j + a − x (1+ i ) t d t + (cid:90) −∞ ( − i ) (cid:0) − ax + ( − i ) t (cid:1) ρ ψ \{ µ j } (cid:0) µ j − ax + ( − i ) t (cid:1) e − xµ j + a − x ( − i ) t d t . Using the substitution s = tx yields that the first integral equals x ρ − e − µ j x (cid:90) ∞ (1 + 2 i ) (cid:0) (1 + i ) s − a (cid:1) ρ j ψ \{ µ j } (cid:0) µ j − ax + (1 + i ) sx (cid:1) e a − (1+ i ) s d s . (28)From dominated convergence the limit of the integral in (28) as x → ∞ is ψ \{ µ j } ( µ j ) (cid:90) ∞ (cid:0) (1 + i ) s − a (cid:1) ρ j e − (1+ i ) s d s . A similar result holds for the second integral. Hence, it has been shown thatlim x →∞ f j ( x ) x ρ j − e − µ j x = ψ \{ µ j } ( µ j ) (cid:90) ∞ (1 + i ) (cid:0) − a + (1 + i ) s (cid:1) ρ j e − ( − a +(1+ i ) s ) d s + ψ \{ µ j } ( µ j ) (cid:90) −∞ ( − i ) (cid:0) − a + ( − i ) s (cid:1) ρ j e − ( − a +( − i ) s ) d s = ψ \{ µ j } ( µ j ) (cid:90) Γ − a z ρ j e − z d z . (29)12 emark 3.1. The starting point of the contour, µ (cid:48) , was set to move righttowards µ j . Another solution could be letting it move left towards µ l (the largestsingularity less than µ j ) with the definition µ (cid:48) = µ l + ax . From redoing all thearguments the following result would be reached: lim x →∞ f j ( x ) x ρ l − e − µ l x = φ ( µ l ) π ( µ l ) (cid:90) Γ a z ρ l e − z d z what appears to be a slower decrease towards 0. However, note that only one ofthe integrals is different from 0: (cid:90) Γ a z − ρ l e − z d z = 0 and (cid:90) Γ − a z − ρ j e − z d z (cid:54) = 0 . Proof of Theorem 3.1.
Assume (cid:96) < (cid:96) > f and f are linear combinations of the f Γ functions. Since x is assumedto be positive all f j ( x ) = 0. Then f ( x ) and f ( x ) are linear combinations of f ( x ) , . . . , f m ( x ) . So in order to study P x ( τ < ∞ ) it is sufficient to determine the behaviour ofthe functions f i ( x ), when x → ∞ . For each each i = 1 , . . . , r there are twopossible situations to consider: α i < α i >
0. It was shown in Lemma 3.1and Lemma 3.2 that either waylim x →∞ f i ( x ) x ρ i − e − µ i x = K i for some constant K i . Since the ruin probability P x ( τ < ∞ ) can be writtenas a linear combination of these functions, the asymptotics are determined bythe function with the slowest decrease. This is f , and since µ is alwaysa singularity for ψ , the exact asymptotic behaviour of f can be found inLemma 3.2.Let the two partial eigenfunctions f and f be the linear combinations f ( x ) = r (cid:88) i =1 c i f i ( x ) and f ( x ) = r (cid:88) i =1 c i f i ( x ) (30)for x >
0. Thenlim x →∞ P x ( τ < ∞ )e − µ x x − pαlλκ − = lim x →∞ f µ ( x )e − p µ x x − pαlλκ − (cid:18) c − f ( (cid:96) ) f ( (cid:96) ) − f ( (cid:96) ) + c f ( (cid:96) ) − f ( (cid:96) ) − f ( (cid:96) ) (cid:19) = K , where K is given by K = (cid:18) ψ \{ µ } ( µ ) (cid:90) Γ − a z pαlλκ e − z d z (cid:19) × (cid:18) c − f ( (cid:96) ) f ( (cid:96) ) − f ( (cid:96) ) + c f ( (cid:96) ) − f ( (cid:96) ) − f ( (cid:96) ) (cid:19) . (31)13ence, the theorem is proved for (cid:96) <
0. With the same arguments for (cid:96) > K = c (cid:18) ψ \{ µ } ( µ ) (cid:90) Γ − a z pαlλκ e − z d z (cid:19) . (32) (cid:96) → −∞ The setup for (cid:96) → −∞ becomes more complicated, since the constants c , . . . , c m and b , . . . , b n in the construction of the partial eigenfunctions depend on (cid:96) . To study P x ( τ ( (cid:96) ) < ∞ ) given by (23) both f i ( x ) and f i ( (cid:96) ), i = 1 ,
2, are needed.For x > (cid:96) < i = 1 the expressions are f ( (cid:96) ) = r − (cid:88) j = − s b j ( (cid:96) ) f j, ( (cid:96) ) f ( x ) = r (cid:88) i =1 c i ( (cid:96) ) f i, ( x ) . This definition excludes the last of the integration contours Γ − s, , . . . , Γ r, . Sim-ilarly, f ( (cid:96) ) and f ( x ) are defined by f ( (cid:96) ) = r (cid:88) j = − s +1 ˜ b j ( (cid:96) ) f j, ( (cid:96) ) f ( x ) = r (cid:88) i =1 ˜ c i ( (cid:96) ) f i, ( x ) , excluding the first of the contours Γ − s, , . . . , Γ r, . The constants c ( (cid:96) ) , . . . , c r ( (cid:96) )and b − s ( (cid:96) ) , . . . , b r − ( (cid:96) ) are found as the solution to a linear equation: . . . N − s ( (cid:96) ) . . . N r − ( (cid:96) )... . . . ... ... . . . ...0 . . . N r Γ − s ( (cid:96) ) . . . N r Γ r − ( (cid:96) ) M . . . M r N − s . . . N r − ... . . . ... ... . . . ... M r Γ . . . M r Γ r N r Γ − s . . . N r Γ r − − M . . . − M r N − s . . . N r − ... . . . ... ... . . . ... − M s Γ . . . − M s Γ r N s Γ − s . . . N s Γ r − c ( (cid:96) )... c r ( (cid:96) ) b − s ( (cid:96) )... b r − ( (cid:96) ) = µ ... µ r , (33)where we denote the first matrix by A ( (cid:96) ). The limit of P x ( τ ( (cid:96) ) < ∞ ) when (cid:96) → −∞ can then be derived. 14 heorem 4.1. The limits c i = lim (cid:96) →−∞ c i ( (cid:96) ) are well defined and non–zerofor i = 1 , . . . , r , and lim (cid:96) →−∞ P x ( τ ( (cid:96) ) < ∞ ) = − r (cid:88) i =1 c i f i, ( x ) . (34) The c i constants are found in the Corollary 4.1 below. Figure 2: Shows lim (cid:96) →−∞ P x ( τ ( (cid:96) ) < ∞ ) as a function of x . Example 4.1.
Assume that r = s = 1 , κ = 1 , p = 2 / , q = 1 / and µ = ν = 1 .Then the limit in (34) is a decreasing function of x as illustrated in Figure 2Proof of Theorem 4.1. Notation: In the proof we will write f ( (cid:96) ) = O ( g ( (cid:96) )) ifthere exists a constant C such that f ( (cid:96) ) ∼ Cg ( (cid:96) ).In the matrix A ( (cid:96) ) only N k Γ j ( (cid:96) ) (for k = 1 , . . . , r and j = − s, . . . , r − (cid:96) . Exploring this dependence by applying the same technique as inthe x → ∞ case yields for k = 1 , . . . , r and i = − s, . . . , − (cid:96) →−∞ N k Γ i ( (cid:96) )e (cid:96)ν − i ( − (cid:96) ) qβ − iλκ − = lim (cid:96) →−∞ (cid:96)ν − i ( − (cid:96) ) qβ − iλκ − (cid:90) Γ i, ψ ( z ) z − µ k e − (cid:96)z d z = ψ \{− ν − i } ( − ν − i ) − ν − i − µ k (cid:90) ˜Γ z − qβ − iλκ e z d z (35)if − ν − i is a zero for ψ . Here˜Γ = { ( − i ) t : 0 ≤ t < ∞} and ψ \{− ν − i } = z − (cid:32) r (cid:89) k =1 ( z − µ k ) − pαkλκ (cid:33) s (cid:89) d =1 ,d (cid:54) = i ( z + ν d ) − qβdλκ . If − ν − i is a singularity the result islim (cid:96) →−∞ N k Γ i ( (cid:96) )e (cid:96)ν − i ( − (cid:96) ) qβ − iλκ − = ψ \{− ν − i } ( − ν − i ) − ν − i − µ k (cid:90) ˜Γ a z − qβ − iλκ e z d z , (36)15here ˜Γ a = { a + (1 + i ) t : −∞ < t ≤ } + { a + ( − i ) t : 0 ≤ t < ∞} . for any a >
0. Furthermorelim (cid:96) →−∞ N k Γ ( (cid:96) ) = ψ \{ } (0) − µ k (cid:90) ˜Γ a z − e z d z . (37)Finally, the constants related to µ , . . . , µ r satisfy the following if µ i is a zerolim (cid:96) →−∞ N i Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pαiλκ = ψ \{ µ i } ( µ i ) (cid:90) ˜Γ z − pαiλκ − e z d z (38)lim (cid:96) →−∞ N k Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pαiλκ − = ψ \{ µ i } ( µ i ) µ i − µ k (cid:90) ˜Γ z − pαiλκ e z d z if k (cid:54) = i (39)and if it is a singularitylim (cid:96) →−∞ N i Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pαiλκ = ψ \{ µ i } ( µ i ) (cid:90) ˜Γ a z − pαiλκ − e z d z (40)lim (cid:96) →−∞ N k Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pαiλκ − = ψ \{ µ i } ( µ i ) µ i − µ k (cid:90) ˜Γ a z − pαiλκ e z d z if k (cid:54) = i . (41)When calculating the determinant of A ( (cid:96) ) it is crucial that N k Γ i ( (cid:96) ) has thelargest rate of growth when k = i . Furthermore, if µ i is a singularity of anorder in (0 ,
1) and k (cid:54) = i then the limit integral for N k Γ i ( (cid:96) ) is zero while theintegral in the limit of N i Γ i ( (cid:96) ) is not. Define the matrices M = M . . . M r N − s . . . N − ... . . . ... ... . . . ... M r Γ . . . M r Γ r N r Γ − s . . . N r Γ − − M . . . − M r N − s . . . N − ... . . . ... ... . . . ... − M s Γ . . . − M s Γ r N s Γ − s . . . N s Γ − and N ( (cid:96) ) = N ( (cid:96) ) . . . N r − ( (cid:96) )... . . . ... N r Γ ( (cid:96) ) . . . N r Γ r − ( (cid:96) ) . The formulas (35) – (41) yield that det( A ( (cid:96) )) ∼ (cid:0) det( N ( (cid:96) ))( − r + s +1 det( M ) (cid:1) and by using that N i Γ i ( (cid:96) ) has the most rapid growth compared to N k Γ i ( (cid:96) ) when k (cid:54) = i , it is seen that det( N ( (cid:96) )) ∼ (cid:32) N r Γ ( (cid:96) ) r − (cid:89) i =1 N i Γ i ( (cid:96) ) (cid:33) N ( (cid:96) )) = O (cid:18) e (cid:96) (cid:80) r − j =1 µ j ( − (cid:96) ) (cid:80) r − j =1 pαjλκ (cid:19) . Cramer’s Rule provides the constants c ( (cid:96) ) , . . . , c r ( (cid:96) ) and b − s ( (cid:96) ) , . . . , b r − ( (cid:96) ) inthe equation system (33): c ( (cid:96) ) = det( A ( (cid:96) ))det( A ( (cid:96) )) , where A ( (cid:96) ) = µ . . . N − s ( (cid:96) ) . . . N r − ( (cid:96) )... ... . . . ... ... . . . ... µ r . . . N r Γ − s ( (cid:96) ) . . . N r Γ r − ( (cid:96) )0 M . . . M r N − s . . . N r − ... ... . . . ... ... . . . ...0 M r Γ . . . M r Γ r N r Γ − s . . . N r Γ r − − M . . . − M r N − s . . . N r − ... ... . . . ... ... . . . ...0 − M s Γ . . . − M s Γ r N s Γ − s . . . N s Γ r − , and similarly for the remaining constants. It is seen thatdet( A i ( (cid:96) )) = O (cid:18) e (cid:96) (cid:80) r − j =1 µ j ( − (cid:96) ) (cid:80) r − j =1 pαjλκ (cid:19) for i = 1 , . . . , r + s and therefore c i ( (cid:96) ) = det( A i ( (cid:96) ))det( A ( (cid:96) )) = O (1) i = 1 , . . . , rb j ( (cid:96) ) = det( A r + s +1+ j ( (cid:96) ))det( A ( (cid:96) )) = O (1) j = − s, . . . , − . Furthermore,det( A r + s +1 ( (cid:96) )) ∼ (cid:32) det( M ) × µ r r − (cid:89) i =1 N i Γ i ( (cid:96) ) (cid:33) det( A r + s +1+ j ( (cid:96) )) ∼ det( M ) × µ j N r Γ ( (cid:96) ) r − (cid:89) i =1 ,i (cid:54) = j N i Γ i ( (cid:96) ) j = 1 , . . . , r − b ( (cid:96) ) = det( A r + s +1 ( (cid:96) ))det( A ( (cid:96) )) ∼ (cid:32) µ r N r Γ ( (cid:96) ) (cid:33) b j ( (cid:96) ) = det( A r + s +1+ j ( (cid:96) ))det( A ( (cid:96) )) ∼ (cid:32) µ j N j Γ j ( (cid:96) ) (cid:33) = O (cid:18) e (cid:96)µ j ( − (cid:96) ) pαjλκ (cid:19) j = 1 , . . . , r − . c ( (cid:96) ) , . . . , ˜ c r ( (cid:96) ) and ˜ b − s +1 ( (cid:96) ) , . . . , ˜ b r ( (cid:96) ) that belongs tothe second partial eigenfunction solve an equation system similar to (33): . . . N − s +1 ( (cid:96) ) . . . N r ( (cid:96) )... . . . ... ... . . . ...0 . . . N r Γ − s +1 ( (cid:96) ) . . . N r Γ r ( (cid:96) ) M . . . M r N − s +1 . . . N r ... . . . ... ... . . . ... M r Γ . . . M r Γ r N r Γ − s +1 . . . N r Γ r − M . . . − M r N − s +1 . . . N r ... . . . ... ... . . . ... − M s Γ . . . − M s Γ r N s Γ − s +1 . . . N s Γ r ˜ c ( (cid:96) )...˜ c r ( (cid:96) )˜ b − s +1 ( (cid:96) )...˜ b r ( (cid:96) ) = µ ... µ r , (42)where the integration contour Γ − s is replaced by Γ r in order to obtain a newand independent partial eigenfunction. It is similarly shown that the constantshave the following asymptotics as functions of (cid:96) ˜ c i ( (cid:96) ) = O (cid:32) µ N ( (cid:96) ) (cid:33) = O (cid:16) e − (cid:96)µ ( − (cid:96) ) pα λκ (cid:17) i = − s, . . . , − b j ( (cid:96) ) = O (cid:32) µ N ( (cid:96) ) (cid:33) = O (cid:16) e − (cid:96)µ ( − (cid:96) ) pα λκ (cid:17) j = − s + 1 , . . . , b j ( (cid:96) ) ∼ (cid:32) µ j N j Γ j ( (cid:96) ) (cid:33) = O (cid:18) e − (cid:96)µ j ( − (cid:96) ) pαjλκ (cid:19) j = 1 , . . . , r . The asymptotic behaviour of the f j, functions is of interest as well. Similarto the previous analysis it is seen that for j = − s, . . . , − (cid:96) →−∞ f j, ( (cid:96) )e (cid:96)ν − j ( − (cid:96) ) qβjλκ − = ψ \{− ν − j } ( − ν − j ) (cid:90) ˜Γ a z − qβjλκ e z d z , if ν − j is a singularitylim (cid:96) →−∞ f j, ( (cid:96) )e (cid:96)ν − j ( − (cid:96) ) qβjλκ − = ψ \{− ν − j } ( − ν − j ) (cid:90) ˜Γ z − qβjλκ e z d z , if ν − j is a root . For j = 0 is lim j →−∞ f , ( (cid:96) ) = ψ \{ } (0) (cid:90) ˜Γ a z − e z d z , and for j = 1 , . . . , r islim (cid:96) →−∞ f j, ( (cid:96) )e − (cid:96)µ j ( − (cid:96) ) pαjλκ − = ψ \{ µ j } ( µ j ) (cid:90) ˜Γ a z − pαjλκ e z d z , if µ j is a singularitylim (cid:96) →−∞ f j, ( (cid:96) )e − (cid:96)µ j ( − (cid:96) ) pαjλκ − = ψ \{ µ j } ( µ j ) (cid:90) ˜Γ z − qαjλκ e z d z , if µ j is a root . By comparing these results with the asymptotics for the constants c i ( (cid:96) ), ˜ c i ( (cid:96) ), b j ( (cid:96) ) and ˜ b j ( (cid:96) ) it is seen that 18 b j ( (cid:96) ) f j, ( (cid:96) ) tends to zero exponentially fast as (cid:96) → −∞ for j = − s, . . . , − • ˜ b j ( (cid:96) ) f j, ( (cid:96) ) tends to zero exponentially fast as (cid:96) → −∞ for j = − s +1 , . . . , • b j ( (cid:96) ) f j, ( (cid:96) ) = O (cid:16) − (cid:96) (cid:17) for (cid:96) → −∞ when j = 1 , . . . , r − • ˜ b j ( (cid:96) ) f j, ( (cid:96) ) = O (cid:16) − (cid:96) (cid:17) for (cid:96) → −∞ when j = 1 , . . . , r .Finally, the non–zero limit of b ( (cid:96) ) f , ( (cid:96) ) when (cid:96) → −∞ islim (cid:96) →−∞ b ( (cid:96) ) f , ( (cid:96) ) = lim (cid:96) →−∞ µ r N r Γ ( (cid:96) ) f , ( (cid:96) )= 1 µ r ψ \{ } (0) (cid:82) ˜Γ a z − e z d z ψ \{ } (0) − µ r (cid:82) ˜Γ a z − e z d z = − . Hence it has been shown thatlim (cid:96) →−∞ f ( (cid:96) ) = lim (cid:96) →−∞ r − (cid:88) j = − s b j ( (cid:96) ) f j, ( (cid:96) ) = − (cid:96) →−∞ f ( (cid:96) ) = lim (cid:96) →−∞ r (cid:88) j = − s +1 ˜ b j ( (cid:96) ) f j, ( (cid:96) ) = 0 . Furthermore it is shown that all ˜ c i ( (cid:96) ) decrease to zero solim (cid:96) →−∞ f ( x ) = lim t →−∞ r (cid:88) i =1 ˜ c i ( (cid:96) ) f i, ( x ) = 0and since all c i has a non–zero limit, then lim (cid:96) →−∞ f ( x ) is well–defined andnon–zero. Thereforelim (cid:96) →−∞ P x ( τ < ∞ )= lim (cid:96) →−∞ f ( x ) 1 − f ( (cid:96) ) f ( (cid:96) ) − f ( (cid:96) ) + f ( x ) f ( (cid:96) ) − f ( (cid:96) ) − f ( (cid:96) ) = − lim (cid:96) →−∞ f ( x ) . The asymptotic expression for c i ( (cid:96) ) can found to be c i ( (cid:96) ) ∼ (cid:32) ( − r + s +1 − i det ( M i ) det ( M ) 1 µ r N r Γ ( (cid:96) ) (cid:33) , M i = M . . . M i − M i +1 . . . M r N − s . . . N ... . . . ... ... . . . ... ... . . . ... M r Γ . . . M r Γ i − M r Γ i +1 . . . M r Γ r N r Γ − s . . . N r Γ − M . . . − M i − − M i +1 . . . − M r N − s . . . N ... . . . ... ... . . . ... ... . . . ... − M s Γ . . . − M s Γ i − − M s Γ i +1 . . . − M s Γ r N s Γ − s . . . N s Γ . Hence we have
Corollary 4.1.
For i = 1 , . . . , r it holds that lim (cid:96) →−∞ c i ( (cid:96) ) = ( − r + s +1 − i det( M i )det( M ) 1 µ r (cid:18) ψ \{ } (0) − µ k (cid:90) ˜Γ a z − e z d z (cid:19) − = ( − r + s − i det( M i )det( M ) (cid:18) ψ \{ } (0) (cid:90) ˜Γ a z − e z d z (cid:19) − . Consider the negative drift case, κ <
0, where the ruin probability is 1. Thissituation is particularly simple because only one partial eigenfunction, f , isneeded, since crossing (cid:96) through continuity is not possible. The Laplace trans-form of the undershoot is therefore expressed by the simple formula E x [e − ζZ ] = f ( x ) . Since ψ satisfies that | ψ ( z ) | = O ( | z | − − λκ ), the negative κ makes infinite in-tegration contours impossible. We shall apply Theorem 2.1 and choose finiteintegration contours as described in [10, Section 5]. However, in [10] the con-tours are suggested to be half–circles and circles, but that choice makes thecalculations of our prblem too complicated. Thus we will use line segmentsinstead. Note that µ is always a zero for ψ . For each i = 2 , . . . , r define:If µ i is a zero define Γ i as { µ i + ( − − i ) t : 0 ≤ t ≤ µ i − µ }∪ { µ i − i ( µ i − µ ) + ( − i ) t : µ i − µ ≤ t ≤ µ i − µ } . If µ i is a singularity define Γ i as { µ i + a − (cid:96) + i ( µ i + a − (cid:96) − µ ) + (1 + i ) t : − ( µ i + a − (cid:96) − µ ) ≤ t ≤ − µ i + a − (cid:96) − µ }∪ { µ i + a − (cid:96) + (1 − i ) t : − µ i + a − (cid:96) − µ ≤ t ≤ }∪ { µ i + a − (cid:96) + ( − − i ) t : µ i + a − (cid:96) − µ ≤ t ≤ }∪ { µ i − a − (cid:96) + i ( µ i + a − (cid:96) − µ ) + ( − i ) t : µ i + a − (cid:96) − µ ≤ t ≤ µ i + a − (cid:96) − µ } . (cid:54) (cid:114) (cid:114) (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) µ i is a zero for ψ . µ µ i Γ i (cid:45)(cid:54) (cid:114) (cid:114) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) µ i is a singularity for ψ . µ µ i µ i + a − (cid:96) Γ i Figure 3: The choice of contours in the negative drift case .A rough sketch of the two contours can be seen on Figure 3. The partialeigenfunction f is defined by f ( y ) = r (cid:88) i =2 c i f Γ i ( y ) + U f ∗ ( y ) + f ( y ) , (43)where f ∗ ( y ) = 1 [ (cid:96) ; ∞ [ ( y ), and the parameters c , . . . , c r and U are the solutionsof the equation − µ ( (cid:96) ) M ( (cid:96) ) · · · M r ( (cid:96) ) − µ ( (cid:96) ) M ( (cid:96) ) · · · M r ( (cid:96) )... ... . . . ... − µ r ( (cid:96) ) M r Γ ( (cid:96) ) · · · M r Γ r ( (cid:96) ) Uc ... c r = − µ + ζ ... − µ r + ζ (44)where we shall denote the first matrix by B ( (cid:96) ) and the constants M k Γ i ( (cid:96) ) aregiven as M k Γ i ( (cid:96) ) = (cid:90) Γ i ψ ( z ) z − µ k e − (cid:96)z d z (45)for i = 2 , . . . , r and k = 1 , . . . , r . To explore the asymptotic behaviour of U, c , . . . , c r and through that the behaviour of f , it is necessary to study theconstants in (45).The following result states that the limit of the undershoot is a simpleexponential distribution with parameter µ from the dominating part of thedownward jumps. Theorem 4.2.
For all ζ ≥ it holds that lim (cid:96) →−∞ E x [e − ζZ ] = µ µ + ζ . Proof.
First the behaviour of the constants M k Γ i ( (cid:96) ) when (cid:96) → −∞ is explored.When µ i is a zero (for some i = 2 , . . . , r ) and i (cid:54) = k the constant can be written21s M k Γ i ( (cid:96) ) = (cid:90) µi − µ ( − − i ) ψ ( µ i + ( − − i ) t ) µ i + ( − − i ) t − µ k e − (cid:96) ( µ i +( − − i ) t ) d t (46)+ (cid:90) µ i − µ µi − µ ( − i ) ψ ( µ i − i ( µ i − µ ) + ( − i ) t ) µ i − i ( µ i − µ ) + ( − i ) t − µ k e − (cid:96) ( µ i − i ( µ i − µ )+( − i ) t ) d t . Rewriting the expression and applying the usual substitution s = − (cid:96)t to thefirst part in (46) yields M k Γ i ( (cid:96) ) = e − (cid:96)µ i ( − (cid:96) ) pλαiκ − (cid:90) ( − (cid:96) ) µi − µ ( − − i ) ψ \{ µ i } ( µ i + ( − − i ) s − (cid:96) ) µ i + ( − − i ) s − (cid:96) − µ k × (( − − i ) s ) − pλαiκ e s ( − − i ) d s . Hence, by dominated convergence it is seen that the integral in the last line hasthe limit ψ \{ µ i } ( µ i ) µ i − µ k (cid:90) ∞ ( − − i )(( − − i ) s ) − pλαiκ e s ( − − i ) d s = ψ \{ µ i } ( µ i ) µ i − µ k (cid:90) − Γ z − pλαiκ e z d z , where − Γ = { ( − − i ) t : 0 ≤ t < ∞} . Now remains to discuss the asymptotics of the second part in (46). Substituting s = − (cid:96) ( t − ( µ i − µ )) the expression equalse − (cid:96) ( µ µi − i µi − µ ) ( − (cid:96) ) − × (cid:82) ( − (cid:96) ) µi − µ ( − i ) ψ (cid:16) µ + µ i − i µ i − µ + ( − i ) s − (cid:96) (cid:17) e s ( − i ) d s . The integral has the following limit for (cid:96) → −∞ ψ (cid:18) µ + µ i − i µ i − µ (cid:19) (cid:90) ˜Γ e z d z by dominated convergence, where ˜Γ = { ( − i ) t : 0 ≤ t < ∞} . Since the firstpart grows with a larger rate than the last part islim (cid:96) →−∞ M k Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pλαiκ − = ψ \{ µ i } ( µ i ) µ i − µ k (cid:90) − Γ z − pλαiκ e z d z . (47)A similar result is found in the case where i = k :lim (cid:96) →−∞ M k Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pλαiκ = ψ \{ µ i } ( µ i ) (cid:90) − Γ z − pλαiκ − e z d z . (48)22he same substitution technique yields results in the cases where µ i are singu-larities for ψ . That giveslim (cid:96) →−∞ M k Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pλαiκ − = ψ \{ µ i } ( µ i ) µ i − µ k (cid:90) − Γ a z − pλαiκ e z d z (49)if i (cid:54) = k and lim (cid:96) →−∞ M k Γ i ( (cid:96) )e − (cid:96)µ i ( − (cid:96) ) − pλαiκ = ψ \{ µ i } ( µ i ) (cid:90) − Γ a z − pλαiκ − e z d z (50)when i = k . Here − Γ a = { a + (1 − i ) t : −∞ < t ≤ } ∪ { a + ( − − i ) t : 0 ≤ t < ∞} . sing (47)-(50) we obtain the following asymptotic behaviour of the determinantof the matrix B ( (cid:96) ), det( B ( (cid:96) )) ∼ (cid:32) − µ r (cid:89) i =2 M i Γ i ( (cid:96) ) (cid:33) . (51)Let B i denote B with the i th column replaced by the vector [ − µ + ζ , . . . , − µ r + ζ ] T ,then det( B ( (cid:96) )) ∼ (cid:32) − µ + ζ r (cid:89) i =2 M i Γ i ( (cid:96) ) (cid:33) (52)det( B i ( (cid:96) )) ∼ (cid:16) − µ µ i + ζ − − µ i µ + ζ (cid:17) (cid:89) j ∈{ ,...,r } ,j = i M j Γ j ( (cid:96) ) . (53)The solutions of equation (44) are obtained from Cramer’s rule, and the asymp-totic behaviour is determined from the results (51)-(53). This yields U ( (cid:96) ) = det( B ( (cid:96) ))det( B ( (cid:96) )) ∼ (cid:32) − µ + ζ − µ (cid:33) = µ µ + ζc i ( (cid:96) ) = det( B i ( (cid:96) ))det( B ( (cid:96) )) ∼ (cid:32) − µ µ i + ζ − − µ i µ + ζ − µ M i Γ i ( (cid:96) ) (cid:33) with i = 2 , . . . , r . Since all M i Γ i ( (cid:96) ) are growing exponentially fast the asymp-totics for f defined in (43) are easily determined, as well as the limit of theLaplace transform for the undershoot,lim (cid:96) →−∞ E x [e − ζZ ] = lim (cid:96) →−∞ (cid:32) r (cid:88) i =2 c i ( (cid:96) ) f Γ i ( x ) + U ( (cid:96) ) f ∗ ( x ) (cid:33) = lim (cid:96) →−∞ U ( (cid:96) ) · µ µ + ζ . eferences [1] Asmussen, S. (2000). Ruin Probabilities World Scientific, Singapore .[2] Borovkov, K and Novikov, A. (2008). On exit times of L´evy–drivenOrnstein–Uhlenbeck processes.
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